An investigation of geometric non-linear formulations for 3D beam elements

This paper presents the Total and Updated Lagrangian formulations for 3D beam elements. Only geometric non-linearity is considered with the assumption of small strains. Performance of 3D beam elements with conventional beam interpolation functions formulated by the approach of Mallett and Marcal (ASCE 94, 2081-2105, 1968) in Total Lagrangian description is studied. The [N1] and [N2] matrices of Mallett and Marcal are derived explicitly by the procedure suggested by Rajasekaran and Murray (ASCE 99, 2423-2437, 1973) under the assumption of small rotations. A simplified updated Lagrangian formulation with equilibrium equations written in terms of relative deformations in convected coordinates based on the work of Oran (ASCE 99, 687-1001, 1973) is investigated for 3D beam elements. Performance of 3D beam element in updated Lagrangian description with equilibrium equations written in terms of total displacements is also investigated. In this formulation, a combination of linear and first order geometric stiffness matrices is used as tangent as well as secant stiffness matrices. Several examples of 2D and 3D framed structures are tested, but only 3D examples are reported in this paper. © 1990.